Chapter 4 Definitions p2

Question 1

Union (A ∪ B)

Answer 1

Union (A ∪ B)
The union of two events A and B, written as A ∪ B, includes all outcomes that are in event A, in event B, or in both. It represents the situation where either A happens, B happens, or both happen. The probability of a union is calculated using the formula:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Unions are often illustrated using Venn diagrams, where the union covers all areas within both circles. This concept is essential for calculating combined probabilities in overlapping events.

Question 2

Mutually Exclusive

Answer 2

Mutually Exclusive
Mutually exclusive events are events that cannot happen at the same time. If one occurs, the other must be impossible. For example, when rolling a die, the events “rolling an even number” and “rolling an odd number” are mutually exclusive. The probability of both happening simultaneously is zero:

P(A ∩ B) = 0

If events are mutually exclusive, the probability of either event happening is simply the sum of their individual probabilities:

P(A ∪ B) = P(A) + P(B)

Question 2

Intersection (A ∩ B)

Answer 2

Intersection (A ∩ B)
The intersection of events A and B, written as A ∩ B, is the set of outcomes that are common to both A and B. It represents the event where both A and B occur simultaneously. In a Venn diagram, the intersection is shown as the overlapping region of the two circles. The probability of the intersection is important for calculating joint events. If A and B are independent events, then the probability of their intersection is given by:

P(A ∩ B) = P(A) × P(B)

Question 3

Complement (A′)

Answer 3

Complement (A′)
The complement of an event A, written as A′ or Ā, includes all outcomes in the sample space that are not in A. It represents the event that A does not occur. The probability of the complement is given by:

P(A′) = 1 − P(A)

Complements are helpful in simplifying probability problems, especially when calculating the probability of “at least one” or “not happening” events. In a Venn diagram, the complement is represented by the area outside the circle for event A.

Question 4

Independent Event

Answer 4

Independent Event
Two events are independent if the occurrence of one does not affect the probability of the other occurring. This means:

P(A ∩ B) = P(A) × P(B)

An example of independent events is tossing a coin and rolling a die—the result of one does not influence the other. Independence is crucial in many probability problems involving repeated or unrelated trials. If events are not independent, different formulas must be applied.

Question 5

Dependent Event

Answer 5

Dependent Event
Dependent events are events where the outcome or occurrence of one event affects the probability of the other. This usually occurs when the events are connected, such as drawing cards from a deck without replacement. For example, if you draw a red card from a deck and do not return it, the probability of drawing another red card changes. The formula used for dependent events is:

P(A ∩ B) = P(A) × P(B | A)

Understanding dependence is essential for accurate conditional probability calculations.

Question 6

Set Notation

Answer 6

Set Notation
Set notation is a formal way of describing collections of objects or outcomes, often written using curly brackets. For example, a set A containing the numbers 2, 4, and 6 is written as:
A = {2, 4, 6}

Sets represent events or groups of outcomes in probability and are typically labelled with capital letters such as A, B, or S (for the sample space). Common symbols in set notation include:

∪ (union)

∩ (intersection)

′ or c (complement)

Elements inside sets are separated by commas, and the symbol ∈ means “is an element of.” Set notation is essential for expressing probabilities and visualising relationships between events, particularly using Venn diagrams.

Question 7

Empty Set

Answer 7

Empty Set
The empty set is a set with no elements, written as ∅ or {}. In probability, it represents an event that cannot happen, meaning its probability is zero. For example, if event A is “rolling a 7 on a standard die,” then:

A = ∅

because that outcome does not exist. In set notation, when two sets have no elements in common, their intersection is the empty set:

A ∩ B = ∅

This concept is useful for identifying mutually exclusive events.